# Linear Differental Equations

A first-order differential equation is a linear equation in the dependent variable $$y$$ if it can be written in the form:

$$a_1(x) \cfrac{dy}{dx} + a_0(x) y = g(x)$$

The standard form of a linear equation is:

$$\cfrac{dy}{dx} + P(x) y = f(x)$$

If $$f(x) = 0$$ the equation is said to be homogeneous and is separable. If the equation is non-homogeneous, we can find an integration factor, $$\mu (x)$$ such that:

$$\cfrac{d}{dx} (\mu (x) y) = \mu (x) (\cfrac{dy}{dx} + P(x) y )$$

$$\cfrac{d \mu}{dx} y + \mu (x) \cfrac{dy}{dx} = \mu (x) \cfrac{dy}{dx} + \mu (x) P(x) y$$

$$\cfrac{d \mu}{dx} y = \mu (x) P(x) y$$

$$\cfrac{d \mu}{dx} = \mu (x) P(x)$$

This equation is separable with solution:

$$\mu (x) = e^ {\int P(x)}$$

Find the integrating factor for the DE $$t y^{'} - 3 y = \frac{\sin {\pi t}}{t}$$

The steps to solve a linear non-homogeneous differential equation are:

1. Write the equation in standard form $$\cfrac{dy}{dx} + P(x) y = f(x)$$.
2. Find the integrating factor $$\mu (x) = e^ {\int P(x)}$$.
3. Multiple both sides by the integrating factor $$\cfrac {d}{dx} (e^ {\int P(x)} y ) = e^ {\int P(x)} f(x)$$.
4. Integrate both sides and isolate for $$y$$.

Solve the DE $$t^3 y^{'} +4t^2 y = e^{-t}$$ subject to $$y(-1) = 0$$.